Problem: All of the 3rd grade teachers and students from Santa Rita went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$39.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$20.00$ each for teachers and $$11.50$ each for students, and the group paid $$140.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 39.5}$ ${20x+11.5y = 140.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-14y = -158}$ ${20x+11.5y = 140.5}$ Add the top and bottom equations together. $ -2.5y = -17.5 $ $ y = \dfrac{-17.5}{-2.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {5x+3.5y = 39.5}$ to find $x$ ${5x + 3.5}{(7)}{= 39.5}$ $5x+24.5 = 39.5$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 7}$ into $ {20x+11.5y = 140.5}$ and get the same answer for $x$ ${20x + 11.5}{(7)}{= 140.5}$ ${x = 3}$ There were $3$ teachers and $7$ students on the field trips.